The city is grilling museums and cultural institutions about the racial and ethnic diversity of their entire staffs — and if they don’t answer, they’ll be cut off from taxpayer cash.

Nearly 1,000 organizations were asked to fill out a survey this summer detailing the race, gender and disability status of their employees and board members.

“As long as you complete the survey, you will be eligible to apply for funding from [the city] in fiscal year 2017,” institutions were told in a July 20 letter from Tom Finkelpearl, commissioner of the Department of Cultural Affairs.

Organizations had to fill out a “demographics spreadsheet” by combing through their personnel files for their workers’ information.

Groups that don’t keep such data were told to hand out voluntary surveys to employees, with the caveat in the instructions that “you cannot require a staff member or board member to self-report any element of their identity.”

In addition to providing the workforce information, the cultural groups had to answer a questionnaire with up to 48 questions — including “How does your organization engage with issues of diversity on a daily basis?” and “What forms of diversity do you think are important for strengthening the quality of work of your organization?”

Groups are also queried on whether there were barriers to increasing diversity among staff and board members, according to a copy of the survey obtained by The Post.

The questionnaire, to be completed online, is being overseen by Ithaka S+R, a Rector Street nonprofit-research service. The forms were to be completed by 6 p.m. on Sept. 18.

The survey is being paid for with more than $150,000 in private funds from the Ford Foundation, the Mertz Gilmore Foundation and the Rockefeller Brothers Fund.

But an art adviser who works with high-level collectors called the racial inquisition another “Big Brother act by the de Blasio ­regime” and said it didn’t matter who was funding the initiative.

“It’s a waste of money that could go into the arts,” the adviser said.

The city plans to hire a “diversity consultant” to coordinate activities coming after the survey. That has led to some worries that particular groups would be punished for not being diverse enough.

But the city says in the survey that it will not know individual ­responses, only the overall figures. The paperwork adds that ­responses will not affect future funding.

The city said most groups have returned the questionnaires but would not give specific figures. It said the initiative has been “widely embraced” by the cultural community.

“A workforce, leadership and audience that reflect our city’s increasingly diverse population is critical for cultural organizations to continue engaging New Yorkers and to strengthen the sector as a whole,” said Ryan Max, a spokesman for the Department of Cultural Affairs.